Jiang, Q., et al.: Influence of Heat In-Leak, Longitudinal Conduction and Property... THERMAL SCIENCE: Year 2019, Vol. 23, No. 3B, pp. 1969-1979 1969 INFLUENCE OF HEAT IN-LEAK, LONGITUDINAL CONDUCTION AND PROPERTY VARIATIONS ON THE PERFORMANCE OF CRYOGENIC PLATE-FIN HEAT EXCHANGERS by Qingfeng JIANG a,b,c, Ming ZHUANG b*, Zhigang ZHU b, Linhai SHENG b, and Ping ZHU b aSchool of Energy and Power, Jiangsu University of Science and Technology, Zhenjiang, Jiangsu, China b Institute of Plasma Physics, Chinese Academy of Sciences, Hefei, Anhui, China c University of Science and Technology of China, Hefei, Anhui, China Original scientific paper https://doi.org/10.2298/TSCI170627235J For helium liquefaction/refrigeration systems, conventional design theory always fails in cryogenic applications and heat exchangers operating at low temperatures are usually sensitive to longitudinal heat conduction, heat in-leak from surroundings and variable fluid properties. Governing equations based on distributed parameter methods are developed to evaluate performance deterioration caused by these effects. The model synthetically considering these loss mechanisms is validated against experimental data and design results obtained by commercial software Aspen MUSETM. Sample multi- stream heat exchangers are further studied to discuss quantitative effects of these heat losses. In accordance with previous researches, the comprehensive effects of various losses are analyzed qualitatively in order to reveal their influences and investigate on the strategies of improving the heat transfer performance. The numerical method is useful in the design procedure of cryogenic heat exchangers and can be adopted to predict heat transfer and pressure drop performance under the actual low temperature environment. Key words: heat exchanger, longitudinal conduction, heat in-leak, helium liquefier, helium refrigerator Introduction Counter-flow plate-fin heat exchangers (PFHE) are commonly utilized in cryogenic applications due to their high effectiveness and compact size. In helium liquefaction/refrigeration systems, PFHE are always used to further cool the coming hot and high pressure helium gas by recuperating most sensible heat from the low pressure vapor. However, for the cryogenic applications, various parameters such as longitudinal heat conduction, heat-in-leak from surroundings and property variations of fluid need to be considered carefully, which are often neglected in the conventional design procedure of heat exchangers. Lumped parameter models, as the basic design theory, i..e. the effectiveness-number of heat transfer units (ε-NTU) method, the logarithmic mean temperature difference method and so on, are based on the assumptions of steady-state, negligible heat transfer from ambient, no longitudinal heat conduction, constant heat capacity rate, constant individual, and overall heat transfer coefficients [1, 2]. When the magnitude of these secondary parameters is too large to be neglected, these closed form solutions would not be precise enough. *Corresponding author, e-mail: [email protected] Jiang, Q., et al.: Influence of Heat In-Leak, Longitudinal Conduction and Property... 1970 THERMAL SCIENCE: Year 2019, Vol. 23, No. 3B, pp. 1969-1979 During the last several decades, these effects have been studied by many researchers based on various different assumptions. For the performance deterioration caused by longitudinal heat conduction, which would flatten the temperature distributions in the wall and thus reduce the effectiveness, Kays and London [1] provided graphical results for the balanced and imbalanced counter-flow heat exchangers. More comprehensive, systematic analytical results assuming constant properties and adiabatic end conditions were proposed by Kroeger [3]. For the condi- tions of the heat exchanger cold end directly connected to the evaporator in Joule-Thomson refrigerators, Narayanan and Venkatarathnam [4] discussed the parasitic heat loss through the wall at the cold end. Hansen et al, [5] investigated the effects of longitudinal conduction and found that there exists an optimal metal conductivity for the geometry considered in the design of heat exchangers incorporated with J-T expansion process. Krishna et al, [6] studied the effect of longitudinal wall conduction on the performance of three-fluid cryogenic heat exchangers. Saberimoghaddam and Abadi [7] found that the higher tube wall thermal conductivity would not bring about a bigger temperature measurement error at tube outlet, while the wall thermal conductivity leaded to longitudinal conduction in the tube wall. In addition, a significant amount of heat in-leak from ambient could be crucial in determining the heat exchanger performance. Chowdhury and Sarangi [8] analyzed a heat exchanger exposed to a parasitic heat transfer that is driven by a temperature difference between a fixed ambient temperature and the local fluid temperature. Gupta and Atrey [9] published the experimental results and numerical modeling considering the heat in-leak from surroundings and longitudinal conduction through the wall. Based on this model, the characterizations of various losses were discussed by Aminuddin and Zubair [10] assuming adiabatic and non-adiabatic end conditions. Different from the assumptions of external heat transfer linearly proportional to the temperature difference between ambient and fluid, Nellis [11] considered radiative heat transfer as the dominant parasitic in a typical cryogenic heat exchanger. The commercial software, Aspen MUSETM [12], has been successfully used to evaluate the performance of PFHE. It can effectively take into accounts the losses due to longitudinal conduction and actual fluid physical properties, but the heat in-leak is empirically represented as a rough input from any source. In order to study these combined effects, many numerical models have been developed. A systematic review of the evolution and challenges in mathematical models of cryogenic heat exchangers was presented by Pacio and Dorao [13]. The differential equations derived by Kroeger [3] under conditions of constant properties could be used to study the effects of longitudinal conduction for balanced and imbalanced flow qualitatively and systematically. Atrey [9] and Gupta et al. [14] extended the model to cover heat in-leak from surroundings. The numerical model presented by Nellis and Gupta et al, [11] considering the effects of property variations, metal conductivity, parasitic heat load, and heat transfer coefficients was useful in the design of a two-stream heat exchangers. To take care of transverse heat conduction for multi- stream heat exchangers, Goyal et al, [15] presented a detailed numerical model with discretizing energy balance equations additionally for fin elements. In this study, the model early proposed is modified at the methodology of analysis and modeling, which can not only reveal the cryogenic loss mechanisms qualitatively, but also analyze the quantitative effects of these losses occurring in multi-stream heat exchangers under the actual cryogenic environment. Based on the model, the predictions are validated against experimental results presented previously. To further examine the flexibility of this model and overcome the insufficiencies of the commercial software, the results about two integrated common cases with special reference to helium liquefaction/refrigeration systems are made comparisons with the predictions. The numerical model could be useful for designers to properly Jiang, Q., et al.: Influence of Heat In-Leak, Longitudinal Conduction and Property... THERMAL SCIENCE: Year 2019, Vol. 23, No. 3B, pp. 1969-1979 1971 forecast the performance deterioration under conditions of considering arbitrary variations of number of transfer units, heat capacity ratios, fluid properties, metal conductivities, parasitic heat loads, heat transfer and frictional resistance coefficients along the length of heat exchangers. Compared with studies reported in previous literatures, qualitative analyses of performance influences caused by longitudinal heat conduction, heat in-leak and property variations are also performed for different number of transfer units and heat capacity rate ratios. Numerical model In this section, governing equations for a counter-flow heat exchanger with external heat transfer to the cold fluid are derived based on the conservation of energy in a control volume and expressed in dimensionless form by commonly used parameters. The computational grid consisting of the necessary nodes and elements is illustrated in fig.1. Considering the effects of longitudinal conduction tend to be concentrated at the ends, an exponentially distributed grid [11], γ = 4. characterized by higher concentration of nodes near the edges is applied. The heat exchanger is divided into n sections. Each section consists of three elements: hot fluid side, cold fluid side and wall boundary. With the following assumptions: Figure 1. Computational grid for the numerical model – Steady-state conditions exist throughout the heat exchanger. – In any cross-section for every passage, the fluid temperature and pressure perpendicular to the flow direction remain unchanged. – The metal matrix temperature inside the core is constant along the width of the heat exchanger. – The fluid in every passage is distributed uniformly. – The governing equations in the ith section are: Hot fluid: Uh,i+Uh,i-1 T +T T +T C +C A’ Δx h,i h,i - 1 ‒T + U A T ‒ h,i h,i - 1 = h,i h,i - 1 (T ‒T ) (1) 2 s,h i ( 2 w ,i ) 0,hi 0,hi ( a 2 ) 2 h,i-1 h,i Wall: Uh,i+Uh,i-1 Th,i+Th,i-1 (kw,i-1 + kw,i) Ac A’s,h Δxi ( ‒ T w ,i ) + (Tw,i-1‒Tw,i) + 2 2 Δxi+Δxi-1 Tw,i-1+Tc,i+1 Uc,i+Uc,i-1 Tc,i+Tc,i+1 (kw,i+kw,i-1) Ac + U0,wi A0,wi (T a ‒ ) = A’s,c Δxi (T w ,i ‒ ) + (Tw,i‒Tw,i+1) (2) 2 2 2 Δxi+Δxi+1 Cold fluid: Uc,i+Uc,i-1 T +T T +T C +C A’ Δx T ‒ c,i c,i - 1 + U A T ‒ c,i c,i - 1 = c,i c,i - 1 (T ‒T ) (3) 2 s,c i ( w ,i 2 ) 0,ci 0,ci ( a 2 ) 2 c,i-1 c,i The energy equations can be analyzed non-dimensionally: Hot fluid: υh,i+ΔXi θh,i+θh,i-1 θh,i+θh,i- 1 μh,i+μh,i-1 ntuh,i-1+ ntuh,i + ‒ θw,i + 2αhiNTUiυhi R‒ +1 = (θh,i- 1‒ θh,i) (4) ( ) ( ) ( ) μh,i υh,i-1+ΔXi-1 2 2 Jiang, Q., et al.: Influence of Heat In-Leak, Longitudinal Conduction and Property..